Every Puzzle Has An Answer
by Frog-kun
Summary: Luke encounters the hideous manifestation known as calculus homework. Layton tells him that there is a way to answer the questions without knowing a thing about calculus. Written for those who hate their maths homework. Complete with hints and solution!
1. The Puzzle

**The Puzzle**

Luke was a clever young boy; most discerning adults could see that. Seeing as he was Layton's apprentice and all, he'd already solved quite a few puzzles that would render most thrice his age snivelling wrecks of their former selves. More than that, though, Luke loved to _learn_: his love of puzzles was matched only by his inquisitiveness (and maybe his love for animals). Whatever he was told, he'd absorb it into his head pretty quickly. It was a bit of a stretch to call him a genius – but clever was certainly the right word for him. As far as he was concerned, Professor Layton was the real, bonafide genius, although Luke did like the sound of 'clever' for himself.

This cleverness persisted, even as Luke got on in years. He went to school (!) and was considered one of the brighter pupils in class. However, since Luke almost never seemed to use what he learned at school in puzzle-solving or in anything that was not related to school at all, for that matter, he had to wonder what point there was in his education. Being Layton's apprentice was so much more educational and useful than some crummy old institution, he thought. (Thus, he chose to ignore the fact that the professor himself was a learned man who had gone to university, among other things.)

The matter came to a head one year when Luke was sixteen and he was taught the epitome of useless things: calculus.

The horrid, vile thing! Luke couldn't bear to even look at his homework and thus kept it perpetually shoved away in the back corner of his desk. He would have set it on fire as well but that wasn't what a _gentleman _did. It wasn't that Luke didn't understand calculus from the teacher's explanation, but it just didn't seem usable to him at all. Frankly, he had much better things to do with his time.

Layton ended up hearing of this the way he always seemed to hear about Luke's problems without ever actively prying. He mentioned it as the two of them had tea together (Luke still liked Oasis Berry; his tastes had never evolved) and had just finished solving a word puzzle in the newspaper.

"I see you have been having quite the trouble with your homework lately, Luke," Layton remarked amiably.

Just the sheer mention of it was enough to wipe away Luke's smile and replace it with a dark scowl.

"It's not trouble, it's simply awfully boring, Professor!"

"Nevertheless," Layton reminded him, "disregarding one's obligations is not a gentlemanly thing to do." Noticing Luke's scowl still hadn't gone away, Layton added, "Now, Luke, why don't you try showing me your homework?"

"Show you, Professor?" Now Luke was confused.

To this, the good professor simply smiled and spread his arms invitingly in that comforting way he always did. "Think of it as a puzzle! Does that make it more manageable?"

"A puzzle..." Luke repeated slowly. He scratched his head. "I... I'll try...!"

Once he had retrieved his homework, he showed it to Layton. Layton read through the sheet quickly enough, and then a smile came upon his face as he neared the end.

"See here," he said, pointing. "This question seems an awful lot like a puzzle, doesn't it?"

Luke read the words aloud: "The sum of two positive integers is 108. Find the numbers such that their product is a maximum." He hesitated. "Er..."

"There is a beauty in mathematics," Layton said, "such that its entire foundations are grounded entirely upon logic and reason. These are precisely the same things we use with puzzles. Maths can be obfuscating, but the logic is nonetheless always present. For instance, there is a way to solve this problem without using any calculus at all."

"You mean...!" Luke was amazed. "You know the answer to the question already?"

"I do," Layton replied. "Now let's see if you can't find the answer yourself."

* * *

**Author's note: **Try to solve this puzzle along with Luke! The next three chapters are hints to the puzzle, and chapter 5 contains the solution. See if you find the answer yourself without looking at too many hints. Read the question carefully. Here it is again in bold:

**The sum of two positive integers is 108. Find the numbers such that their product is a maximum.**

You can probably hear the Layton puzzle theme ringing in your ears as you read this, can't you?


	2. Hint One

**Hint One**

Layton is right in saying you can solve this puzzle without calculus. In fact, you don't even need to use algebra at all. That said, deciphering mathematical language can be tricky. Let's try and figure out what the question is asking you:

Integers mean whole numbers such as 1, 2, 3, etc. Fractions and decimals won't be part of your answer. 'Sum' means two numbers added together. 'Product' means two numbers multiplied together.

The question is asking you to find a pair of numbers that add up to 108 which, when multiplied together, give the highest total out of any other pair.


	3. Hint Two

**Hint Two**

There are many pairs of numbers that add to give 108. You aren't using combinations that have a high number in them, are you? Then you'll end up having to multiply that number with a small one...


	4. Hint Three

**Hint Three**

The maximum is 2916. If your two numbers multiply to give this as the answer, then you've found the solution.


	5. The Solution

**The Solution**

Both numbers are 54. Their product is 2916, which is the highest possible given that the numbers have to add up to 108.

In these types of questions, the highest possible product comes from the two numbers which are closest together. The further apart the numbers are, the lower the value becomes. Once you pick up on this trend, the puzzle becomes relatively simple.

For those who were curious as to how calculus is used, here is how a mathematician would solve this puzzle:

Let x = one number.  
The sum of the two numbers is 108.  
∴ 108 – x is the other number.  
P (the product) = x(108 – x)  
P = 108x – x^2  
∴ P' = 108 – 2x  
Let P' = 0  
108 – 2x = 0  
∴ 2x = 108  
∴ x = 54  
x = 54 so the first number is 54.  
108 – x is the second number so it is also 54.  
∴ 54 and 54 are the numbers required to produce a maximum product.

The answer is calculated by finding the gradient derivative, which can be calculated by using the first principles of differentiation. When the gradient derivative, that is, when f'(x) = 0, the function being described is at a stationary point, which usually takes the form of a local maximum, minimum or a point of inflection. By finding the x-values which make f'(x) = 0, mathematicians can calculate what numbers correspond with the maximum.

Sounds awfully convoluted, doesn't it?

* * *

"See?" said Layton. "It wasn't too difficult, was it?"

"Not at all!" Luke seemed rather stunned at how quickly he himself had come upon the answer. "Professor, I hope you'll do my homework for me in the future!"

Layton laughed. "I hoped you'd come to see maths as another outlet for puzzles."

"Oh," said Luke. He was _very _disappointed, to say the least.

Still, he'd always known the professor would not be such a good teacher if he took it upon himself to do all the solving. This was all part of Luke's broader education.

And maybe... _maybe_... it had made him see maths in a somewhat different light.

* * *

_Epilogue_

"Professor! I've got another one!" Luke said brightly the next time he and Layton met. "This one's about trigonometry!"

**In a moment of foolishness, a student decides to take a bungee jump. On the way down, he reflects that his height in metres above the water at any time **_**t **_**seconds after leaping will be described by the function h(t) = 20cos(0.6t) + 25. If the cord is perfectly elastic, after how many seconds will he be back to the level of the platform?**

"A moment of foolishness indeed," remarked the professor.

"You can help me solve it without using trigonometry, can't you, professor?" Luke asked.

Layton took a moment to contemplate his answer.

"I say, Luke, this reminds me of an absolutely smashing puzzle about bungee jumping I once heard. Would you like to hear it?"


End file.
